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How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?

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  • $\begingroup$ Does $H_n$ mean the n-th harmonic number? Perhaps the tag ({tag:harmonic-numbers]) is also applicable, if yes. $\endgroup$ – Martin Sleziak Dec 12 '13 at 15:58
  • $\begingroup$ Yes, n-th harmonic number. Thanks for the notice. $\endgroup$ – user95733 Dec 12 '13 at 16:09
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    $\begingroup$ There are very many similar questions linked on the right, under Related. Perhaps you can find some inspiration from the answers provided there ? $\endgroup$ – Lucian Dec 12 '13 at 16:14
  • $\begingroup$ See a related problem.you need to modify it a little to solve your problem. $\endgroup$ – Mhenni Benghorbal Dec 19 '13 at 14:14
  • $\begingroup$ See this question. The Op posted a nice formula that you can use. $\endgroup$ – Mhenni Benghorbal Dec 19 '13 at 14:22
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Let's start with the product of $\;-\ln(1-x)\,$ and $\dfrac 1{1-x}$ to get the product generating function
(for $|x|<1$) : $$\tag{1}f(x):=-\frac {\ln(1-x)}{1-x}=\sum_{n=1}^\infty H_n\, x^n$$ Dividing by $x$ and integrating we get : \begin{align} \sum_{n=1}^\infty \frac{H_n}n\, x^n&=\int \frac{f(x)}xdx\\ &=-\int \frac{\ln(1-x)}{1-x}dx-\int\frac{\ln(1-x)}xdx\\ \tag{2}&=C+\frac 12\ln(1-x)^2+\operatorname{Li}_2(x)\\ \end{align} (with $C=0$ from $x=0$)
The first integral was obtained by integration by parts, the second from the integral definition of the dilogarithm or the recurrence for the polylogarihm (with $\;\operatorname{Li}_1(x)=-\ln(1-x)$) : $$\tag{3}\operatorname{Li}_{s+1}(x)=\int\frac {\operatorname{Li}_{s}(x)}x dx$$

Dividing $(2)$ by $x$ and integrating again returns (using $(3)$ again) : \begin{align} \sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n&=\int \frac {\ln(1-x)^2}{2\,x}dx+\int \frac{\operatorname{Li}_2(x)}x dx\\ &=C+I(x)+\operatorname{Li}_3(x)\\ \end{align} with $I(x)$ obtained by integration by parts (since $\frac d{dx}\operatorname{Li}_2(1-x)=\dfrac {\ln(x)}{1-x}$) : \begin{align} I(x)&:=\int \frac {\ln(1-x)^2}{2\,x}dx\\ &=\left.\frac{\ln(1-x)^2\ln(x)}{2}\right|+\int \ln(1-x)\frac {\ln(x)}{1-x}dx\\ &=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)\right|+\int \frac{\operatorname{Li}_2(1-x)}{1-x}dx\\ &=\left.\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)-\operatorname{Li}_3(1-x)\right|\\ \end{align} getting the general relation : $$\tag{4}\sum_{n=1}^\infty \frac{H_n}{n^2}\, x^n=C+\frac{\ln(1-x)^2\ln(x)}{2}+\ln(1-x)\operatorname{Li}_2(1-x)+\operatorname{Li}_3(x)-\operatorname{Li}_3(1-x)$$ (with $C=\operatorname{Li}_3(1)=\zeta(3)$ here)
applied to $x=\dfrac 12$ with $\operatorname{Li}_2\left(\frac 12\right)=\dfrac{\zeta(2)-\ln(2)^2}2$ from the link returns the wished : \begin{align} \sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\frac{\ln(2)^3}2-\ln(2)\frac{\zeta(2)-\ln(2)^2}2\\ \tag{5}\sum_{n=1}^\infty \frac{H_n}{n^2\;2^n}&=\zeta(3)-\ln(2)\frac{\zeta(2)}2 \end{align}

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We will make frequent use of $$ \binom{n+1}{k+1}=\binom{n}{k}\frac{n+1}{k+1}\tag{1} $$ The Generalized Harmonic Numbers of the second order are defined as $$ H_n^{(2)}=\sum_{k=1}^n\frac1{k^2}\tag{2} $$ The factor of $2^{-n}$ in each term reminded me of the Euler Series Transformation. Reversing the series acceleration (series deceleration?), it can be seen that we should look at $$ \begin{align} \sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1} &=\sum_{k=0}^n\binom{n}{k}\frac{(-1)^k}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{3}\\ &=\frac1{n+1}\sum_{k=0}^n(-1)^k\binom{n+1}{k+1}\sum_{j=0}^k\frac1{(j+1)^2}\tag{4}\\ &=\frac1{n+1}\sum_{j=0}^n\sum_{k=j}^n(-1)^k\binom{n+1}{k+1}\frac1{(j+1)^2}\tag{5}\\ &=\frac1{n+1}\sum_{j=0}^n(-1)^j\binom{n}{j}\frac1{(j+1)^2}\tag{6}\\ &=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\binom{n+1}{j+1}\frac1{j+1}\tag{7}\\ &=\frac1{(n+1)^2}\sum_{j=0}^n(-1)^j\sum_{k=0}^n\binom{k}{j}\frac1{j+1}\tag{8}\\ &=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\sum_{j=0}^n(-1)^j\binom{k+1}{j+1}\tag{9}\\ &=\frac1{(n+1)^2}\sum_{k=0}^n\frac1{k+1}\tag{10}\\[3pt] &=\frac{H_{n+1}}{(n+1)^2}\tag{11} \end{align} $$ Explanation:
$\ \;(3)$: use $(2)$
$\ \;(4)$: apply $(1)$
$\ \;(5)$: change order of summation
$\ \;(6)$: $\sum\limits_{j=k}^n(-1)^j\binom{n+1}{j+1}=(-1)^k\binom{n}{k}$
$\ \;(7)$: apply $(1)$
$\ \;(8)$: $\sum\limits_{j=0}^n\binom{j}{k}=\binom{n+1}{k+1}$
$\ \;(9)$: $\sum\limits_{j=0}^k(-1)^j\binom{k+1}{j+1}=1$
$(10)$: $H_{n+1}=\sum\limits_{k=0}^n\frac1{k+1}$

Using $(11)$, the Euler Series Transformation says that $$ \begin{align} \sum_{k=0}^\infty(-1)^k\frac{H_{k+1}^{(2)}}{k+1} &=\sum_{n=0}^\infty2^{-n-1}\sum_{k=0}^n\binom{n}{k}(-1)^k\frac{H_{k+1}^{(2)}}{k+1}\\ &=\sum_{n=0}^\infty2^{-n-1}\frac{H_{n+1}}{(n+1)^2}\tag{12} \end{align} $$ Therefore, with a change of indexing, we get $$ \begin{align} \sum_{n=1}^\infty2^{-n}\frac{H_n}{n^2} &=\sum_{n=1}^\infty(-1)^{n-1}\frac{H_n^{(2)}}{n}\tag{13}\\ &=\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^3}+\sum_{n=1}^\infty(-1)^{n-1}\frac{H_{n-1}^{(2)}}{n}\tag{14}\\ &=\frac34\zeta(3)+\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n}\sum_{k=1}^{n-1}\frac1{k^2}\tag{15}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=k+1}^\infty\frac{(-1)^{n-1}}{nk^2}\tag{16}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty\frac{(-1)^{k+n-1}}{(k+n)k^2}\tag{17}\\ &=\frac34\zeta(3)+\sum_{k=1}^\infty\sum_{n=1}^\infty(-1)^{k+n-1}\left(\frac1{k^2n}-\frac1{kn(k+n)}\right)\tag{18}\\[6pt] &=\frac34\zeta(3)-\frac12\zeta(2)\log(2)+\frac14\zeta(3)\tag{19}\\[9pt] &=\zeta(3)-\frac12\zeta(2)\log(2)\tag{20} \end{align} $$ Explanation:
$(13)$: reindex $(12)$
$(14)$: $H_n^{(2)}=\frac1{n^3}+H_{n-1}^{(2)}$
$(15)$: apply $(2)$
$(16)$: change order of summation
$(17)$: reindex $n\mapsto k+n$
$(18)$: $\frac1{(k+n)k^2}=\frac1{k^2n}-\frac1{kn(k+n)}$
$(19)$: $\sum\limits_{k=1}^\infty\sum\limits_{n=1}^\infty\frac{(-1)^{k+n}}{kn(k+n)}=\frac14\zeta(3)$ from $(5)$ and $(7)$ of this answer
$(20)$: addition

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  • $\begingroup$ +1. Nice, Rob! The factor of $2^n$ in the denominator made me think of the Euler series transformation, too, but I couldn't make it go through. Also, your work starting from equation (13) is an evaluation of $A(2,1)$ from my question here, which means you've got a derivation for the last of the three sums from that question. I'd be happy if you were willing to finish off your series of answers to my question with your derivation for $A(2,1)$ here. $\endgroup$ – Mike Spivey Dec 17 '13 at 3:52
  • $\begingroup$ @MikeSpivey: I was going to do so, but dinner interrupted. Note that the justification for $(19)$ cites my answer to your question. $\endgroup$ – robjohn Dec 17 '13 at 5:12
  • $\begingroup$ @MikeSpivey: I have now completed the third answer to your question. $\endgroup$ – robjohn Dec 17 '13 at 8:59
  • $\begingroup$ Nice answer robjohn. $\endgroup$ – Mathsource Jul 23 '14 at 6:35
  • $\begingroup$ While $(9)$ is still true, the index should be till $n$ (in the explanation part) as per what you used in the answer. $\endgroup$ – MathGod Jun 17 '17 at 2:47
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\begin{eqnarray} \sum\limits_{n=1}^\infty \frac{H_n}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \sum\limits_{n=m}^\infty \frac{1}{n^2 2^n} = \sum\limits_{m=1}^\infty \frac{1}{m} \int\limits_{-\infty}^0(-\xi) \frac{(1/2 \exp(\xi))^m}{1-1/2 \exp(\xi)} d\xi = \\ \int\limits_{-\infty}^0 \xi \frac{\log(1 - 1/2 \exp(\xi))}{1-1/2 \exp(\xi)} d\xi = \\ \int\limits_{1/2}^1 \left(\frac{1}{u} + \frac{1}{1-u}\right) \log(u) \left[\log(2) + \log(1-u)\right] du = \\ \zeta(3) - \frac{1}{12} \pi^2 \log(2) \end{eqnarray}

I think that all the steps are clear except for the last two ones.In the second last step I substituted for 1 - 1/2 exp(xi).The only non-trivial integrals in here are $\int \log(u)/(1-u) du$ and $\int \log(u) \log(1-u)/(1-u) du$. I compute them now. The first integral is done by expanding the denominator in a series and integrating term by term. \begin{equation} \int\limits_{1/2}^1 \frac{\log(u)}{1-u} du = \sum\limits_{p=0}^\infty \int\limits_{1/2}^1 u^p \log u du = \sum\limits_{p=0}^\infty \frac{-1+2^{-1-p}}{(p+1)^2} + \log(2) \sum\limits_{p=0}^\infty \frac{2^{-1-p}}{p+1} = -\zeta(2) + Li_2(1/2) + \log(2) Li_1(1/2) \end{equation} The second integral is done by integrating by parts and using the definition of the polylogarithmic function. \begin{equation} \int\limits_{1/2}^1 \log(u) \frac{\log(1-u)}{1-u} du = \int\limits_{1/2}^1 \log(1-u) Li_2^{'}(1-u) du = \left.\log(1-u) Li_2(1-u)\right|_{1/2}^1 + Li_3(1/2) = \log(2) Li_2(1/2) + Li_3(1/2) \end{equation} Now, the only thing that remains is to bring the results together. I am sorry but due to time constraints I am not able to do it right now. I have verified with Mathematica that all the partial results are correct.

Final Note: We can clearly see that the result is expressed though elementary functions and through polylogarithms of order not bigger than three, evaluated at 1/2.From the Wikipedia page on Polylogarithms we learn those polylogarithms at 1/2 are expressed in closed form through $\pi$, $\log(2)$ and the $\zeta$ functions. Having said that we can say that this completes the proof.

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    $\begingroup$ Unfortunately, I would describe this as cheating more than slightly :( $\endgroup$ – Igor Rivin Dec 12 '13 at 17:16
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Here is a different approach without using the generating function proved by Raymond Manzoni

The following identity holds :$\displaystyle \sum_{n=1}^{\infty}\frac{H_n}{2^n n^2}=\zeta(3)-\frac{1}{2}\ln(2)\zeta(2)$

Proof: Using $ \displaystyle \frac{\ln(1-x)}{1-x}=-\displaystyle \sum_{n=1}^{\infty}H_n x^n $

multiply both sides by $ \ln x/x $ then integrate from $ x=0 $ to $ 1/2 $ \begin{align*} I&= \int_0^{1/2}\frac{\ln x\ln(1-x)}{x(1-x)}\,dx=-\sum_{n=1}^{\infty}H_n \int_0^{1/2} x^{n-1}\ln x\ dx\\ &=-\sum_{n=1}^\infty H_n\frac{\partial}{\partial{n}} \int_0^{1/2} x^{n-1}=-\sum_{n=1}^\infty H_n\frac{\partial}{\partial{n}} \left( \frac{1}{2^n n}\right) \\ &=-\sum_{n=1}^\infty H_n\left( \frac{\ln2}{2^n n}+\frac{1}{2^n n^2}\right)=-\ln2\sum_{n=1}^\infty\frac{H_n}{2^n n}-\sum_{n=1}^\infty \frac{H_n}{2^n n^2} \tag{1} \end{align*} on the other hand \begin{equation*} I= \int_{0}^{1/2}\frac{\ln x\ln(1-x)}{x(1-x)}\ dx \overset{x\mapsto1-x}{=} \int_{1/2}^{1}\frac{\ln(1-x) \ln(x)}{x(1-x)}\ dx \end{equation*} and by adding the integral to both sides, we get \begin{equation*} 2I= \int_{0}^{1}\frac{\ln x\ln(1-x)}{x(1-x)}\ dx =-\sum_{n=1}^\infty H_n \int_0^1 x^{n-1}\ln x \ dx =\sum_{n=1}^\infty \frac{H_n}{n^2} \end{equation*} substituting the value of $\displaystyle \sum_{n=1}^\infty \frac{H_n}{n^2}=2\zeta(2)$ yields \begin{equation*} I=\zeta(3)\tag{2} \end{equation*}

combining $(1)$ and $(2)$ gives the closed form where we used the generating function

$\displaystyle \sum_{n=1}^\infty\frac{x^n H_n}{n}=\operatorname{Li_2}(x)+\frac12 \ln^2(1-x)$ for $ x=1/2 $ to find $\displaystyle \sum_{n=1}^\infty\frac{H_n}{2^n n}=\frac12\zeta(2)$

note that we used the Dilogarithm reflection formula $ \operatorname{Li_2}(x)+\operatorname{Li_2}(1-x)=\zeta(2)-\ln x\ln(1-x) $ for, $ x\in(1,-1) $, to find $ \operatorname{Li_2}\left( \frac12\right) =\frac12\zeta(2)-\frac12\ln^22 $ where we set $ x=1/2 $

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